Introduction to Graphing

Cartesian Coordinate System

The Cartesian Coordinate System is a fundamental tool in algebra for representing points, equations, and geometric figures in a two-dimensional plane. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0, 0).

• Quadrants: The plane is divided into four quadrants, labeled I, II, III, and IV, starting from the upper right and moving counterclockwise.

• Points: Each point is represented as an ordered pair (x, y), where x is the horizontal position and y is the vertical position relative to the origin.

Example: Plotting points (−2, 4), (3, 3), and (0, −3) involves locating each on the plane according to their x and y values.

Solutions of Equations

A solution of an equation is a value or set of values for the variable(s) that make the equation true. In the context of graphing, solutions often correspond to points on the graph of the equation.

• Verification: Substitute the values into the equation to check if the equality holds.

Example: For the equation :

• Test (−5, 7): (Not a solution)

• Test : (Solution)

Graphs of Equations

To graph an equation means to draw the set of all points (x, y) that satisfy the equation. The graph visually represents the relationship between variables.

• x-intercept: The point where the graph crosses the x-axis, found by setting y = 0.

• y-intercept: The point where the graph crosses the y-axis, found by setting x = 0.

Example: For :

• x-intercept: ; point is (9, 0).

• y-intercept: ; point is (0, 6).

The Distance Formula

The Distance Formula calculates the length between two points and in the plane.

• Formula:

Example:

• Between (−2, 2) and (3, −6):

• Between (−1, −5) and (−1, 2):

Midpoint of Segments

The Midpoint Formula finds the point exactly halfway between two endpoints and .

• Formula:

Example: For endpoints (−4, −2) and (2, 5):

• Midpoint:

Circles

A circle is the set of all points in a plane that are a fixed distance (radius) from a given point (center). The equation of a circle describes this set algebraically.

• Standard Form: where (h, k) is the center and r is the radius.

• Circle at Origin:

Example:

• Circle with radius 5 and center (3, −7):

• Circle : Center is (2, −1), radius is

Summary Table: Key Formulas

Additional info: Academic context and explanations have been expanded for clarity and completeness, including definitions, formulas, and examples for each concept.